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An Estimate of Canonical Dimension of Groups Based on Schubert Calculus

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An Estimate of Canonical Dimension of Groups Based on Schubert Calculus An estimate of canonical dimension of groups based on Schubert calculus Rostislav Devyatov∗ November 3, 2020 Abstract We sketch the proof of a connection between the canonical (0-)dimension of semisimple split simply connected groups and cohomology of their full flag varieties. Using this connection, we get a new estimate of the canonical (0-)dimension of simply connected split exceptional groups of type E understood as a group. A full proof will be published later. 1 Introduction To define the canonical (0-)dimension of an algebraic group understood as a group, we first need to define the canonical (0-)dimension of a scheme understood as a scheme (which is a different definition). Roughly speaking, the canonical (0-)dimension of a scheme is a number indicating how hard it is to get a rational point in the scheme. The canonical (0-)dimension of an algebraic group shows how hard it is to get rational points in torsors related to the group. To be more precise, let us fix some conventions and give some definitions. We speak of algebraic schemes and use stacks project as the source of basic definitions. All schemes in the present text are of finite type over a field and separated. The base field is arbitrary. Speaking of canonical dimension of schemes, there are two closely related notions in the literature: the canonical 0-dimension of a scheme defined in [14] and the canonical dimension of a scheme defined in [9]. These two definitions are not known to be always equivalent, but they are equivalent for two particular classes of schemes: for smooth complete schemes and for torsors of split reductive groups (see [13, Theorem 1.16, Remark 1.17, and Example 1.18]). The definition from [14] looks more motivated, so we are going to use it. Definition 1.1 ([14, Section 4a, first paragraph of Section 4b, and the last paragraph of Section 2a]). Given a scheme X over a field K, the canonical 0-dimension of X understood as a scheme (notation: cd0(X)) is: cd0(X)= max min trdegK L0. 0 0 arXiv:2011.00950v1 [math.AG] 2 Nov 2020 L=a field containing K L =a subfield of L, K⊆L XL has a rational point XL0 still has a rational point A bit less formally, canonical dimension can be explained as follows. Suppose we have expanded the base field K to L, and got a rational point in XL. How large can L be, compared to K? In general, it can be very large, this is unbounded. A related question with a finite answer is: how many algebraically independent generators do we have to keep, at worst (for the worst L), to still have a rational point after scalar expansion (not necessarily the same rational point that we found after expanding scalars to L)? This number of generators is the canonical dimension of X. For more properties of canonical dimension, see [14] in the case of general X and [8] in the case of smooth projective X. We have underlined above that we want to get a rational point over a field between K and L, but not necessarily the same rational point. If we demanded to get the same rational point, we would get ∗Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany. Email address: [email protected] 1 the definition of the essential 0-dimension of a scheme, which is known to coincide with the (standard in algebraic geometry) dimension, see [13, Proposition 1.2]. This can be viewed as a motivation for the word “dimension”. (But essential dimension is not only defined for schemes, and in broader generality it becomes a much more nontrivial notion.) Another motivation for canonical (0-)dimension comes from incompressible varieties, but this moti- vation is only valid for the canonical (0-)dimension of smooth complete schemes. The definition that we are going to give next, the canonical 0-dimension of an algebraic group, and that will be used in the main theorem of this text, does not involve the canonical (0-)dimension of smooth complete schemes, so this motivation will be useless for us. One can find details for this motivation in [8, Section 2]. The second object we need to define before we can define the canonical 0-dimension of a group is a torsor of a group. All algebraic groups in this text are affine. All reductive, semisimple, and simple groups in this text are smooth. Torsors of algebraic groups (over a point) are, informally speaking, homogeneous spaces that are “as large as the group itself”. This notion is mostly interesting over non algebraically closed fields. Definition 1.2 ([14, Section 3a]). Given an algebraic group G, a G-torsor over a point (or simply a G-torsor) is a scheme E with an action ϕ: G × E → E such that (ϕ, pr2): G × E → E × E, where pr2 is the projection to the second factor, is an isomorphism. It is known that all torsors of affine algebraic groups over a point are affine. Finally, the canonical 0-dimension of an algebraic group understood as a group measures how hard it is to get rational points in torsors, informally speaking, related to the group. Precisely: Definition 1.3 ([14, Section 4g]). given an algebraic group G over a field F , the canonical 0-dimension of G understood as a group (notation: cd0(G)) is cd0(G)= max max cd0(E). K=a field containing F E=a GK -torsor The definition of canonical dimension of an algebraic group understood as a group in [9, Introduction] repeats this definition almost exactly, with the only difference being that instead of cd0(E) it uses the definition of canonical dimension of E understood as a scheme from the paper [9] itself. But as we already mentioned above, it is known that these two notions are known to be equivalent for torsors of split reductive groups. So, Definition 1.3 is also equivalent to the definition of canonical dimension of a group from [9, Introduction] for split reductive groups. All groups whose canonical dimension we are going to estimate in this text are split reductive (and even simply connected semisimple), so these results also estimate the canonical dimension in the sense of [9, Introduction]. To formulate the main goal of this text precisely, we need to introduce some more notation and terminology. Given a split semisimple algebraic group G and a Borel subgroup B, the corresponding Weyl group W , and the element w0 ∈ W of maximal length, for each w ∈ W we denote the Schubert −1 variety Bw0w B/B ⊆ G/B by Zw. This Zw is a Schubert divisor if and only if w is a simple reflection, and we denote all Schubert divisors by D1,...,Dr. It is known that the classes [Zw] ∈ CH(G/B) for all w ∈ W form a free set of generators of n1 nr CH(G/B) as of an abelian group. We say that a product of classes of Schubert divisors [D1] ... [Dr] is multiplicity-free if there exists w ∈ W such that the coefficient at [Zw] in the decomposition of n1 nr [D1] ... [Dr] into a linear combination of Schubert classes equals 1. Now we can formulate the goal of this text precisely. Our goal is to sketch the proof of the following theorem. Theorem 1.4. Let G be a split semisimple simply connected algebraic group over an arbitrary field, let B be a Borel subgroup, let r be the rank of G, and let D1,...,Dr ⊂ G/B be the Schubert divisors n1 nr corresponding to the r simple roots of G. If [D1] ... [Dr] is a multiplicity-free product of Schubert divisors, then cd0(G) ≤ dim(G/B) − n1 − ... − nr. As a corollary of this theorem and [5, Theorem 11.5], we will immediately get the following: Corollary 1.5. Let G be a split semisimple simply connected algebraic group of type Er. Then cd0(G) ≤ 17, 37, or 86 for r =6, 7, or 8, respectively. 2 The canonical dimension of simply connected split groups of type Ar and Cr is known to be zero. For types Br and Dr, the canonical dimension was estimated (and computed exactly if r is a power of 2) by N. Karpenko in [10]. The paper [10] also relates cohomology of flag varieties of orthogonal groups (more precisely, orthogonal Grassmannians, not full flag varieties) to canonical dimension, and in this text, we are going to follow the ideas of several proofs from [10]. For type G2, the canonical dimension (of a split simply connected group) is known and equals 3, see [1, Example 10.7]. For type F4, no nontrivial upper bounds on the canonical dimension are known. In types Er, the most difficult part of obtaining Corollary 1.5 was to understand which products of Schubert divisors are multiplicity-free (and this was understood in [5] by the author). The part of the argument establishing relation between Schubert calculus and canonical dimension (in other words, the proof of Theorem 1.4 itself) was known to the experts (or at least they believed that the argument is doable this way). However, we were unable to find an exposition suitable for more general mathematical audience. The present paper contains such an exposition. Acknowledgments I thank Kirill Zaynoulline for bringing my attention to the problem. I thank Nikita Karpenko for useful discussions and explanations about theory of torsors and canonical dimension and about his paper [10]. I also thank Vladimir Chernousov and Alexander Merkurjev for useful discussions about torsors and theory of Galois descent.
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